Dirichlet and Neumann problems for Laplace and heat equations in domains with right angles

The Dirichlet and Neumann problems are considered in the $n$-dimensional cube and in a right angle. The right-hand side is assumed to be bounded, and the boundary conditions are assumed to be zero. We obtain a priori bounds for solutions in the Zygmund space, which is wider than the Lipschitz space $C^{1,1}$ but narrower that the Hölder space $C^{1,\alpha}$, $0<\alpha<1$. Also, the first and second boundary problems are considered for the heat equation with similar conditions. It is shown that the solutions belongs to the corresponding Zygmund space.